Subspaces with extra invariance nearest to observed data.

CARLOS CABRELLI; CAROLINA MOSQUERA

Rio de Janeiro

Congreso; 30 Coloquio Brasileiro de Matematica; 2015

IMPA

Shift invariant spaces are closed subspaces of $L^2(\mathbb{R}^d)$ that are invariant under integer translations. These spaces can also have extra invariance, that is they could be invariant under translates other than integers. Such spaces are important in applications specially in those where the error in approximations is an issue. In this talk, given an arbitrary finite data $F= \{f_1, \dots, f_m\}\subset L^2(\mathbb{R}^d),$ we prove the existence and show how to construct a ``small shift invariant space'' $V$ that is the ``closest'' to the data $F$ over certain class of closed subspaces of $L^2(\mathbb{R}^d).$ The approximating subspace is required to have extra-invariance properties, that is to be invariant under translations by a prefixed additive subgroup of $\mathbb{R}^d$ containing $\mathbb{Z}^d.$Here small means that our solution subspace should be generated by the integer translates of small number of generators.We give an expression for the error in terms of the data and construct a Parseval frame for the optimal space.We also consider the problem of approximating $F$ from generalized Paley-Wiener spaces of $\mathbb{R}^d,$ that are generated by the integer translates of finite number of functions. These spaces can be seen as finitely generated shift invariant spaces that are $\mathbb{R}^d$ invariant. We show the relations between these spaces and multi-tile sets of $\mathbb{R}^d,$ and the connections with recent results on Riesz basis of exponentials.The results are based on a joint work with Carlos Cabrelli.